Predicting What Comes Next: Sequences and Progressions | EOT

Question 7

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the $2^{\text{nd}}$ hour, $4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

Solution

The number of bacteria grows by multiplying by 2 each hour.

Step 1 — Understand the growth pattern

We start with 30 bacteria. The number of bacteria doubles every hour. Let's find the number of bacteria after 1 hour.

$\text{Bacteria after 1 hour} = 30 \times 2^1$

$= 30 \times 2$

$\boxed{60 \text{ bacteria}}$

This shows the pattern of multiplication.

Step 2 — Bacteria at the 2nd hour

We need to find the number of bacteria after 2 hours. We multiply the initial number by 2 for each hour.

$\text{Bacteria after 2 hours} = 30 \times 2^2$

$= 30 \times 4$

$\boxed{120 \text{ bacteria}}$

Step 3 — Bacteria at the 4th hour

Now, let's find the number of bacteria after 4 hours. We use the same pattern for 4 hours.

$\text{Bacteria after 4 hours} = 30 \times 2^4$

$= 30 \times 16$

$\boxed{480 \text{ bacteria}}$

Step 4 — Bacteria at the nth hour

Finally, we find a general formula for n hours. The exponent matches the number of hours passed.

$\text{Bacteria after n hours} = 30 \times 2^n$

$\boxed{30 \times 2^n \text{ bacteria}}$

Answer

(i) At the end of the 2nd hour = 120 bacteria (ii) At the end of the 4th hour = 480 bacteria (iii) At the end of the nth hour = $30 \times 2^n$ bacteria

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Q14

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